NARS is a reasoning system designed under the Assumption of Insufficient Knowledge and Resources (AIKR).
The logic implemented in NARS is NAL (Non-Axiomatic Logic), which will be introduced in multiple layers. At each layer, the grammar rules, semantic definitions, and inference rules of the logic will be further extended, to increase the expressive power of the language and the inferential power of the logic.
For each NAL-n, an idealized version of it is defined first, which is a binary axiomatic logic, called IL-n (Inheritance Logic, the n-th layer).
What is needed is a logic where the notion of "evidence" can be naturally defined. However, as shown by Hempel's Confirmation Paradox (a.k.a. Raven Paradox), FOL runs into trouble here.
The logics in IL-NAL family all belong to the "term logic" tradition, which use subject-predicate sentences and syllogistic rules (while the predicate logic tradition use predicate-argument sentences and truth-functional rules).
A term is an internal identifier in NARS, and its simplest form is a word in a given alphabet.
In an inheritance statement, a subject term and a predicate term are linked together by a copula called "inheritance". Inheritance is defined by being reflexive and transitive, and interpreted as the generalization-specialization relation.
The language of IL-1 contains inheritance statements as sentences.
A non-empty and finite set of statements in IL-1, K, can be treated as the experience for a system using the logic.
The single inference rule of IL-1 corresponds to the transitivity of inheritance. The transitive closure of K is the system's knowledge, K*.
Given K, a statement is true if it is either in K* or is a tautology, otherwise it is false.
IL-1 accepts Closed-World Assumption.
Given K, the extension of a term includes its known specializations; its intension includes its known generalizations. The two together form the meaning of the term.
The above definitions of truth-value and meaning form an Experience-Grounded Semantics (EGS), which is very different from Model-theoretic semantics, while similar to Proof-theoretic semantics.
The system implementing IL-1 can answer simple questions by searching its knowledge.
The key point is to define evidence for an inheritance statement.
Theorem: an inheritance statement is equivalent to the inclusion of the extension of the subject in the extension of the predicate, as well as to the inclusion of the intension of the predicate in the intension of the subject.
Therefore, an inheritance statement summarizes many pairs of inheritance statements, each of which provides a piece of evidence.
This definition of evidence includes Nicod's Criterion as a special case.
NAL uses an idealized experience in IL to define its semantic notions, while the actual experience of the system contains Narsese sentences.
Each judgment in the system has an evidential base in the system's experience.
The Confirmation Paradox does not appear here, because the Equivalence Condition is no longer held in NAL-1.
Wason's Selection Task can be re-analyzed similarly. The common response is not a fallacy if truth-value depends on both positive and negative evidence.
Popper's proposed asymmetry between falsification and verification can be criticized in the same way: on its assumption that a "theory" is logically a universally quantified proposition.
NAL-1 also achieves a unification in the representation of uncertainty: randomness comes from extension; fuzziness comes from intension; ignorance comes from the future.
It is not easy to revise predicate logic for this purpose, while to do it in a term logic is easy and natural, given the subject-copula-predicate structure.